Standard Deviation Calculator
Population standard deviation and variance.
Inputs
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Formula
σ = √(Σ(xᵢ − μ)² ÷ N), where μ is the mean and N is the count
This is the population standard deviation — each value's squared deviation from the mean is averaged over all N values, then square-rooted. Variance (σ²) is that average squared deviation before the square root.
Worked example
For 2, 4, 4, 4, and 5 the mean μ is 3.8. Squared deviations sum to 4.8, giving variance σ² = 0.96 and population standard deviation σ ≈ 0.9798.
Where this can give the wrong answer
- At least two values are required — with a single number there's no spread to measure, so the calculator returns an error.
- This uses the population formula (divide by N), not the sample formula (divide by N − 1) — spreadsheet STDEV.S will give a slightly larger number for the same data.
- Identical values produce σ = 0 — if every number is the same, there's zero variation.
FAQ
- Population (σ, divide by N). Use it when your five inputs are the entire group you care about. If they're a sample meant to represent a larger population, many textbooks and Excel's STDEV.S use N − 1 instead — this calculator does not.